Estimating the scale parameter of a Lomax distribution by using Maximum-Likelihood estimator

Question

I recently came across the Lomax distribution and tried to fit the parameters to some data I have. To do so, I wanted to use the Maximum-Likelihood method. The pdf of the Lomax distribution is given by $f(x) = ab(1+bx)^{-(a+1)}\mathbb{I}(x\geq 0)$. I tried to estimate both $a$ and $b$ via the Maximum-Likelihood method. For $a$ it is easy to see that the ML-estimator is given by $\frac{n}{\sum_{i=1}^n log(1+bx_i)}=a$. Unfortunately, the log-likelihood is monoton increasing in $b$. How can I estimate this parameter?